Nice find!
That recoil starter is a retrofit right enough, though it's still a Pope item, as you've been told. See the Engine Parts List Part 1, in the post CyberJack has linked above.
I've found some Trove ads, that should help with narrowing down the timeframe for this one.
First one is an excerpt from a Waltons NSW ad from 1961, which I'd say would refer to either the 320-07 or 320-09:
![[Linked Image]](https://www.outdoorking-forum.com.au/forum/uploads/usergals/2015/09/full-5630-24302-waltons_mower_ad_1961_11_29.png)
Source: The Biz, Fairfield NSW, 29 Nov 1961, p.4.
LINK Second one is from nearly a year later, and clearly refers to the 657-1-9 and 657-1-11 engine type/s:
![[Linked Image]](https://www.outdoorking-forum.com.au/forum/uploads/usergals/2015/09/full-5630-24306-youngs_pope_ad_1962_10_19.png)
Source: The Canberra Times, ACT, 19 Oct 1962, p.9.
LINKThe distinguishing features are the cowling/fuel tank configuration, and that the earlier 502-01-700 and 502-02-700 engines look to have had plain main bearings, while the 657 series 4-strokes had a ball bearing on the flywheel side, and needle roller on the PTO side. This is touted as the 'new model' in the ad.